Connection of CTMC process, deterministic and stochastic differential equations in modeling of epidemics

Abstract

Mathematical modeling has been used in many fields of study including in epidemiology. The main objective of this study is to show the connection of three mathematical models often used to study the dynamics of disease spread in the natural world; i.e., a stochastic process (CTMC), deterministic model (ODEs) and stochastic differential equation (SDEs). We show that, by proper scaling technique, it is possible to derive the deterministic analogue of a CTMC. Its stochastic differential equation (SDE) version can be obtained by adding a white noise or Weinner process in the deterministic model with proper means and covariance. We demonstrate all three models with the dynamics of SIR epidemics followed by several numerical experiments to show how accurate the trajectories of ODEs follow the sample paths of both CTMC and SDEs.